Geometrization, torsion, and mathematical physics

Geometry lies at the foundation of our understanding of the natural world. Modern research into geometry builds on the lessons of millennia to develop the language and tools used to describe a wide range of phenomena across every scientific discipline. This project addresses specific open questions in geometry whose resolution will require further development of novel general-purpose tools which will deepen our understanding of this central topic. Furthermore, this project will support the training of personnel in the most cutting-edge geometric ideas and methods. On a more technical level, the project will take inspiration from various developments in topology, mathematical physics, and complex geometry which suggest natural notions of Ricci curvature, Einstein metrics, and Ricci flow, which incorporate torsion. The project will uncover fundamental geometric and analytic properties of these structures, which entails generalizing some of the most fundamental parts of our understanding of modern geometry, including: establishing an extension of Gromov-Ruh's Almost flat manifolds theorem, extending the Calabi-Yau theorem to more general settings in complex geometry, developing the analysis of the physical Hull-Strominger system, and developing an existence and uniqueness theory for geometric structures on hyperbolic three-manifolds which extends the Mostow rigidity theorem. This project will reveal new structural links between the various subjects touched by Ricci curvature and will result in the development of broadly applicable insights into analysis, partial differential equations, differential and complex geometry, mathematical physics, and topology. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.